Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m×n matrix A and its singular value decomposition:

$A=U\Sigma V^T\$

First, each matrix $A \in \mathbf{R}^{m \times n}$ (A has m rows and n columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace	definition	containing space	dimension	basis
column space, range or image	im(A) or range(A)	$\mathbf{R}^m$	r (rank)	The first r columns of $\mathbf{U}$
nullspace or kernel	ker(A) or null(A)	$\mathbf{R}^n$	n − r (nullity)	The last (n − r) columns of $\mathbf{V}$
row space or coimage	im(AT) or range(AT)	$\mathbf{R}^n$	r (rank)	The first r rows of $\mathbf{V}^T$
left nullspace or cokernel	ker(AT) or null(AT)	$\mathbf{R}^m$	m − r (corank)	The last (m − r) rows of $\mathbf{U}^T$

Secondly:

1. In $\mathbf{R}^n$, $\mathrm{ker}(A) = (\mathrm{im}(A^T))^\perp$, that is, the nullspace is the orthogonal complement of the row space
2. In $\mathbf{R}^m$, $\mathrm{ker}(A^T) = (\mathrm{im}(A))^\perp$, that is, the left nullspace is the orthogonal complement of the column space.

The four subspaces associated to a matrix A.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.

Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as $A\colon V \to W$ and $A^* \colon W^* \to V^*$: the kernel and image of A ^* are the cokernel and coimage of A.

See also

• Rank–nullity theorem
• Closed range theorem

References

• Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.
• Strang, Gilbert (1993), "The fundamental theorem of linear algebra", American Mathematical Monthly 100 (9): 848–855, doi:10.2307/2324660, JSTOR 2324660, http://www.eng.iastate.edu/~julied/classes/CE570/Notes/strangpaper.pdf 

External links

• Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces at Google Video, from MIT OpenCourseWare